Two bar magnets are in line with each other with their north and south poles near but not touching and they are secured in place. A third magnet is placed in the center air gap but perpendicular. Lets assume the magnets have a strength of 10k gauss. The center magnet will be repelled by one and attracted to the other depending on which pole is placed near. Would the attracting and repelling forces be additive or would the center magnet become saturated and only be able to sustain 10k gauss?
To a good approximation, the magnetic field inside the magnets remains unchanged, even in the presence of other magnets. The forces are additive.